4. Divide \( -x^{3}-3 x^{2}-4 x+3 \) by \( x+2 \). \( \begin{array}{ll}\text { a. }-x^{2}-x-2 \\ \text { b. }-x^{2}-5 x-6 & \text { c. }-x^{2}-5 x-6, \mathrm{R}-1 \\ \text { d. }-x^{2}-x-2, \mathrm{R} 7\end{array} \) Divide using synthetic division. \( \begin{array}{ll}\text { 5. Divide } 4 x^{3}+16 x^{2}+17 x+15 \text { by }(x+3) . & \text { c. }-4 x^{2}-28 x-29, \mathrm{R}-30 \\ \text { a. }-4 x^{2}-4 x-5 & \text { d. } 4 x^{2}+28 x+29, \mathrm{R} 30 \\ \text { b. } 4 x^{2}+4 x+5 & \text { (. }\end{array} \)

Solve the equation by following steps: - step0: Solve for \(x\): \(4x^{3}+16x^{2}+17x+15=0\) - step1: Factor the expression: \(\left(x+3\right)\left(4x^{2}+4x+5\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x+3=0\\&4x^{2}+4x+5=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=-3\\&x \notin \mathbb{R}\end{align}\) - step4: Find the union: \(x=-3\) Solve the equation \( -x^{3}-3 x^{2}-4 x+3=0 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve for \(x\) in the complex numbers system: \(-x^{3}-3x^{2}-4x+3=0\) - step1: Calculate: \(\begin{align}&x\approx 0.51598\\&x\approx -1.75799+1.650348i\\&x\approx -1.75799-1.650348i\end{align}\) - step2: Rewrite: \(x_{1}\approx -1.75799-1.650348i,x_{2}\approx -1.75799+1.650348i,x_{3}\approx 0.51598\) - step3: Remove the complex number(s): \(x\approx 0.51598\) Let's solve the problems step by step. ### Problem 4: Divide \( -x^{3}-3 x^{2}-4 x+3 \) by \( x+2 \) using synthetic division. 1. **Set up synthetic division**: - The divisor is \( x + 2 \), so we use \( -2 \) for synthetic division. - The coefficients of the polynomial \( -x^{3}-3 x^{2}-4 x+3 \) are \( -1, -3, -4, 3 \). 2. **Perform synthetic division**: - Write down the coefficients: \[ \begin{array}{r|rrrr} -2 & -1 & -3 & -4 & 3 \\ & & & & \\ \end{array} \] - Bring down the first coefficient: \[ \begin{array}{r|rrrr} -2 & -1 & -3 & -4 & 3 \\ & & & & \\ & -1 & & & \\ \end{array} \] - Multiply \( -2 \) by \( -1 \) and add to the next coefficient: \[ \begin{array}{r|rrrr} -2 & -1 & -3 & -4 & 3 \\ & & 2 & & \\ & -1 & -1 & & \\ \end{array} \] - Repeat the process: - Multiply \( -2 \) by \( -1 \) and add to \( -4 \): \[ \begin{array}{r|rrrr} -2 & -1 & -3 & -4 & 3 \\ & & 2 & 2 & \\ & -1 & -1 & -2 & \\ \end{array} \] - Multiply \( -2 \) by \( -2 \) and add to \( 3 \): \[ \begin{array}{r|rrrr} -2 & -1 & -3 & -4 & 3 \\ & & 2 & 2 & 6 \\ & -1 & -1 & -2 & 0 \\ \end{array} \] 3. **Result**: - The quotient is \( -x^{2} - x - 2 \) and the remainder is \( 0 \). - Therefore, the answer is \( -x^{2} - x - 2 \). ### Problem 5: Divide \( 4 x^{3}+16 x^{2}+17 x+15 \) by \( x+3 \) using synthetic division. 1. **Set up synthetic division**: - The divisor is \( x + 3 \), so we use \( -3 \) for synthetic division. - The coefficients of the polynomial \( 4 x^{3}+16 x^{2}+17 x+15 \) are \( 4, 16, 17, 15 \). 2. **Perform synthetic division**: - Write down the coefficients: \[ \begin{array}{r|rrrr} -3 & 4 & 16 & 17 & 15 \\ & & & & \\ \end{array} \] - Bring down the first coefficient: \[ \begin{array}{r|rrrr} -3 & 4 & 16 & 17 & 15 \\ & & & & \\ & 4 & & & \\ \end{array} \] - Multiply \( -3 \) by \( 4 \) and add to the next coefficient: \[ \begin{array}{r|rrrr} -3 & 4 & 16 & 17 & 15 \\ & & -12 & & \\ & 4 & 4 & & \\ \end{array} \] - Repeat the process: - Multiply \( -3 \) by \( 4 \) and add to \( 17 \): \[ \begin{array}{r|rrrr} -3 & 4 & 16 & 17 & 15 \\ & & -12 & -12 & \\ & 4 & 4 & 5 & \\ \end{array} \] - Multiply \( -3 \) by \( 5 \) and add to \( 15 \): \[ \begin{array}{r|rrrr} -3 & 4 & 16 & 17 & 15 \\ & & -12 & -12 & -15 \\ & 4 & 4 & 5 & 0 \\ \end{array} \] 3. **Result**: - The quotient is \( 4x^{2} + 4x + 5 \) and the remainder is \( 0 \). - Therefore, the answer is \( 4x^{2} + 4x + 5 \). ### Final Answers: - For Problem 4: \( -x^{2} - x - 2 \) - For Problem 5: \( 4x^{2} + 4x + 5 \)